New we show \(u^*\)  defined in (REF ) is an optimal feedback control of problem (REF ).
Let \(X^*(t)\)  be the associated solution of (REF ) with \(u\)  replaced by \(u^*\) . By Theorem 4.2 in [1]},
\(P_{1}(0,i_0)(x^+)^2+P_{2}(0,i_0)(x^-)^2&=\inf _{u\in \mathcal {U}} J_{T}(x,i_0,u(\cdot ))= J_{T}(x,i_0,u^*(\cdot ))\\&=\mathbb {E}\bigg \lbrace \int _0^T\Big (Q(t, \alpha _t)X^*(t)^2+u^*(t)^{\prime }R(t, \alpha _t)u^*(t)\Big )dt\\&\qquad \qquad +P_{1}(T,\alpha _T)(X^*(T)^+)^2+P_{2}(T,\alpha _T)(X^*(T)^-)^2\bigg \rbrace \\&\ge \mathbb {E}\int _0^T\Big (Q(t, \alpha _t)X^*(t)^2+u^*(t)^{\prime }R(t, \alpha _t)u^*(t)\Big )dt.\) 
